Question: Solve for $x$ : $ 2|x + 10| + 6 = 5|x + 10| + 2 $
Subtract $ {2|x + 10|} $ from both sides: $ \begin{eqnarray} 2|x + 10| + 6 &=& 5|x + 10| + 2 \\ \\ {- 2|x + 10|} && {- 2|x + 10|} \\ \\ 6 &=& 3|x + 10| + 2 \end{eqnarray} $ Subtract $2$ from both sides: $ \begin{eqnarray} 6 &=& 3|x + 10| + 2 \\ \\ {- 2} && {- 2} \\ \\ 4 &=& 3|x + 10| \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{4} {{3}} = \dfrac{3|x + 10|} {{3}} $ Simplify: $ \dfrac{4}{3} = |x + 10| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -\dfrac{4}{3} = x + 10 $ or $ \dfrac{4}{3} = x + 10 $ Solve for the solution where $x + 10$ is negative: $ - \dfrac{4}{3} = x + 10$ Subtract ${10}$ from both sides: $ \begin{eqnarray} - \dfrac{4}{3} &=& x + 10 \\ \\ {- 10} && {- 10} \\ \\ -\dfrac{4}{3} - 10 &=& x \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $3$ $ - \dfrac{4}{3} {- \dfrac{30}{3}} = x $ $ -\dfrac{34}{3} = x $ Then calculate the solution where $x + 10$ is positive: $ \dfrac{4}{3} = x + 10 $ Subtract ${10}$ from both sides: $ \begin{eqnarray} \dfrac{4}{3} &=& x + 10 \\ \\ {- 10} && {- 10} \\ \\ \dfrac{4}{3} - 10 &=& x \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $3$ $ \dfrac{4}{3} {- \dfrac{30}{3}} = x $ $ -\dfrac{26}{3} = x $ Thus, the correct answer is $x = -\dfrac{34}{3} $ or $x = -\dfrac{26}{3} $.